Some Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums
© Zheng and Fu; licensee Springer 2012
Received: 3 August 2012
Accepted: 14 December 2012
Published: 28 December 2012
Some new generalized Volterra-Fredholm type nonlinear discrete inequalities involving four iterated infinite sums are established in this paper. To illustrate the validity of the established inequalities, we present some applications for them, in which new explicit bounds for the solutions of certain infinite sum-difference equations are deduced.
In recent years, many researchers have focused on various generalizations of the known Gronwall-Bellman inequality [1, 2], which provide explicit bounds for unknown solutions of certain difference equations, and a lot of such generalized inequalities have been established in the literature [3–20] including the known Ou-Iang inequality . In , Ma generalized the discrete version of Ou-Iang’s inequality in two variables to a Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of the solutions of certain Volterra-Fredholm type difference equations. But since then, few results on Volterra-Fredholm type discrete inequalities have been established. Recent results in this direction include the works of Zheng , Ma , Zheng and Feng  to our best knowledge. We notice that the Volterra-Fredholm type discrete inequalities in [22–24] are constructed by an explicit function in the left-hand side (see [, Theorems 2.5, 2.6], [, Theorems 2.1, 2.5, 2.6, 2.7], [, Theorems 5, 8, 10, 11]).
Motivated by the works in [22–24], in this paper, we establish some new generalized Volterra-Fredholm type discrete inequalities involving four iterated infinite sums with the right-hand side denoted by an arbitrary function , which are of more general forms. To illustrate the usefulness of the established results, we also present some applications for them and study the boundedness of the solutions of certain Volterra-Fredholm type infinite sum-difference equations.
Throughout this paper, ℝ denotes the set of real numbers and , and ℤ denotes the set of integers, while denotes the set of nonnegative integers. In the next of this paper, let , where , and let be two constants. If U is a lattice, then we denote the set of all ℝ-valued functions on U by and denote the set of all -valued functions on U by . Finally, for a function , we have provided .
2 Main results
Lemma 2.1 [, Lemma 2.1]
Since is selected from Ω arbitrarily, then substituting with in (12), we get the desired inequality (3). □
where is defined in (16).
Combining (19), (22) and (23), we get the desired result. □
The proof for Corollary 2.4 can be completed by setting , , , in Theorem 2.3.
Combining (34), (36) and (37), we get the desired result. □
Combining (46), (48) and (49), we get the desired result. □
The proof for Theorem 2.7 is similar to the combination of Theorem 2.5 and Theorem 2.6, and we omit the details here.
Remark 2.8 We note that the inequalities established in Theorems 2.3, 2.5-2.7 are essentially different from the results in [22–24] as the left-hand side of the inequalities established here is an arbitrary function . Furthermore, if we set , , then Theorem 2.5 reduces to [, Theorem 2.5].
In this section, we present some applications for the results established above. Similar to the applications in [22–24], we research a certain Volterra-Fredholm sum-difference equation and derive some new bounds for its solutions.
where , is an odd number, , .
Combining (53)-(55), we can deduce the desired result. □
Combining (58)-(60), we can deduce the desired result. □
The authors thank the referees very much for their careful comments and valuable suggestions on this paper.
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